The Incompressible Non-Relativistic Navier-Stokes Equation from Gravity (0810.1545v3)

Abstract: We note that the equations of relativistic hydrodynamics reduce to the incompressible Navier-Stokes equations in a particular scaling limit. In this limit boundary metric fluctuations of the underlying relativistic system turn into a forcing function identical to the action of a background electromagnetic field on the effectively charged fluid. We demonstrate that special conformal symmetries of the parent relativistic theory descend to `accelerated boost' symmetries of the Navier-Stokes equations, uncovering a possibly new conformal symmetry structure of these equations. Applying our scaling limit to holographically induced fluid dynamics, we find gravity dual descriptions of an arbitrary solution of the forced non-relativistic incompressible Navier-Stokes equations. In the holographic context we also find a simple forced steady state shear solution to the Navier-Stokes equations, and demonstrate that this solution turns unstable at high enough Reynolds numbers, indicating a possible eventual transition to turbulence.

A Formal Analysis of the Incompressible Non-Relativistic Navier-Stokes Equation from Gravity

This paper by Bhattacharyya, Minwalla, and Wadia explores the intriguing connection between gravity and fluid dynamics through the AdS/CFT correspondence. Their research focuses specifically on the relationship between relativistic hydrodynamics and the incompressible non-relativistic Navier-Stokes equations. Here, we delve into the academic dissection of their findings, highlighting technical results and their implications.

The authors demonstrate how equations of relativistic hydrodynamics reduce to the incompressible Navier-Stokes equations under a specific scaling limit characterized by large distances, long times, low speeds, and low amplitudes. They achieve this reduction while maintaining consistency with the well-known behavior of non-relativistic fluid flows which become effectively incompressible at speeds significantly lower than the speed of sound. This scaling approach uncovers a symmetry structure descended from the special conformal symmetries of the parent relativistic theory, revealing accelerated boost symmetries of the Navier-Stokes equations.

Their investigation shows that the non-relativistic incompressible Navier-Stokes equations can be interpreted within a holographic framework, thereby establishing a gravity dual description for these fluid dynamics. This interpretation uses insights derived from the AdS/CFT correspondence whereby solutions to Einstein's equations in an asymptotically locally AdS space correspond to incompressible Navier-Stokes equations on the boundary.

Numerical Results

The paper rigorously analyses the Navier-Stokes equations using specific boundary conditions and forcing functions. For instance, the authors employ a gauge field $A_x = \alpha e^{i(\omega_0 t + k_0 y)} +$ cc, to derive steady-state solutions under which the fluid velocity exhibits spatial dependencies. Through these analyses, they ascertain that at high Reynolds numbers, the fluid flow becomes unstable, suggesting a transition towards turbulence.

Theoretical and Practical Implications

From a theoretical standpoint, this work enriches the understanding of fluid dynamics within a gravitational context. It advances the prospect that classical gravity in asymptotically AdS spaces can offer fresh perspectives on fluid behaviors like turbulence—a phenomenon notoriously challenging to predict or model accurately in high Reynolds number regimes.

Practically, these insights into fluid dynamics could influence computational fluid dynamics simulations and theoretical models aimed at complex fluid scenarios, possibly aiding in more accurate predictions in engineering and meteorological applications.

Future Directions

This paper hints at potential advances in AI, particularly in computational models that leverage holographic principles for simulating fluid behaviors. Future research may focus on exploring turbulent flows within this framework and developing computational tools that harness holographically dual descriptions for practical fluid dynamics applications.

In conclusion, the authors make a notable contribution to the field by bridging concepts from gravity and fluid dynamics, inviting further exploration into the symbiotic relationships between different physical theories and their practical utilities.